This post introduces a practical view of learning nonlinear dynamics using Koopman-inspired deep linear operators, with a focus on prediction and control.
Why care about learning nonlinear systems?
Problem setup
Consider the discrete-time dynamical system
$$ \boldsymbol{x}(t+1) = \boldsymbol{f}(\boldsymbol{x}(t), \boldsymbol{u}(t)) \label{eq:dynamics} $$where $t=0,1,2,\cdots$ denotes the discrete-time index, $\boldsymbol{x}(t) \in \mathbb{R}^n$ is the system state, $\boldsymbol{u}(t) \in \mathbb{R}^m$ is the control input, and $\boldsymbol{f}:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}^n$ is the system dynamics mapping funciton.
Suppose we are given a dataset
$$ \mathcal{D} = \{(\boldsymbol{x}_i, \boldsymbol{u}_i, \boldsymbol{x}_i^+)\}_{i=1}^{N} \label{eq:dataset} $$where $\boldsymbol{x}_i^+$ denotes the successor state obtained by applying the input $\boldsymbol{u}_i$ to $\boldsymbol{f}$ at $\boldsymbol{x}_i$. Here, the subscript is used to index data samples in the dataset, rather than the time-varying system variables. The problem of interest is to learn an approximation of $\boldsymbol{f}$ from the dataset $\mathcal{D}$.
What is linear operator and state-of-the-art methods
State-of-the-art methods
Popular baseline methods:
- Linear regression works well when $\boldsymbol{f}$ is linear. Common variants include Ordinary Least Squares (OLS) for basic modeling and regularized least squares such as Ridge regression.
- Multilayer Perceptron (MLP) in machine learning: MLP assumes that the unknown dynamics $\boldsymbol{f}$ in $\eqref{eq:dynamics}$ can be represented by a parameterized function class, denoted as $\boldsymbol{f}(\boldsymbol{x}(t), \boldsymbol{u}(t))=\boldsymbol{\phi}(\boldsymbol{x}(t), \boldsymbol{u}(t), \boldsymbol{w}^*)$, where $\boldsymbol{\phi}:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}^n$. The dynamics learning problem is then formulated as estimating the optimal parameters $\boldsymbol{w}^*$ that best fit the dataset $\mathcal{D}$ in $\eqref{eq:dataset}$ by minimizing the prediction error:
One can typically use gradient descent to solve $\eqref{eq:mlp-loss}$
$$ \boldsymbol{w}(k+1) = \boldsymbol{w}(k) - \alpha_w(k) \nabla_{\boldsymbol{w}}\mathbf{L}(\boldsymbol{w}(k)) \label{eq:gd-update} $$where $k=0,1,2,\cdots$ denotes the iteration index, $\alpha_w(k)$ is the learning rate, and $\nabla_{\boldsymbol{w}}\mathbf{L}(\boldsymbol{w}_k)$ denotes the gradient of $\mathbf{L}$ with respect to $\boldsymbol{w}$ evaluated at $\boldsymbol{w}(k)$.
Some notes:
- The assumption that $\boldsymbol{f}$ can be represented by $\boldsymbol{\phi}$ (i.e., the existence of $\boldsymbol{w}^*$) is generally unclear in practice and depends on the expressive power of the chosen function class.
- $\boldsymbol{\hat w}^*$ denotes the optimal parameters with respect to the training dataset, while $\boldsymbol{w}^*$ denotes the optimal parameters defined over the entire state–input space of the dynamical system. In general, $\boldsymbol{\hat w}^*$ and $\boldsymbol{w}^*$ are not equal. The gap between $\boldsymbol{w}^*$ and $\boldsymbol{\hat w}^*$ may lead to poor deployment performance of $\boldsymbol{\hat w}^*$ for state–input pairs that are not contained in the training dataset.
- One useful existing result for $\eqref{eq:gd-update}$ is that, to achieve $\lim_{k\rightarrow\infty} \parallel \boldsymbol{w}(k) - \boldsymbol{\hat w}^*\parallel^2 = 0$, the learning rate $\alpha_w(k)$ should be diminishing. Furthermore, for a constant $\alpha_w$, $\lim_{k\rightarrow\infty} \parallel \boldsymbol{w}(k) - \boldsymbol{\hat w}^*\parallel^2$ typically converges to a small constant.
Linear operator viewpoint
The Koopman operator provides a way to study nonlinear dynamics through a linear evolution of observables. Instead of evolving the state directly, it evolves carefully chosen functions of the state, which can make prediction and control design more structured.
The Koopman-operator-based methods include:
- Dynamic Mode Decomposition (DMD)
- Extended Dynamic Mode Decomposition (EDMD)
- Deep Koopman Operator
The Proposed Method
Main ideas
A typical training objective combines one-step prediction, multi-step rollout, reconstruction, and regularization losses.
Algorithm
- Learn an encoder $\phi_\theta$ that maps original states to latent observables.
- Fit linear operators $(A, B)$ in latent space from data.
- Learn a decoder $\psi_\eta$ to reconstruct physical states from latent variables.
- Train end-to-end with losses on one-step prediction, multi-step rollout, and reconstruction.
Applications
Deep linear-operator models support multiple downstream tasks:
- Long-horizon forecasting in nonlinear physical systems.
- System identification when first-principles models are unavailable or expensive.
- Controller synthesis in latent space using linear optimal control methods.
- Safety filtering and constrained control by combining latent predictions with CBF/MPC layers.
- Model-based reinforcement learning with improved planning and sample efficiency.
Extensions
Current research directions include:
- Time-varying and input-dependent operators $(A_t, B_t)$.
- Stochastic and uncertainty-aware Koopman models for noisy environments.
- Physics-informed lifting functions that embed invariants or conservation structure.
- Distributed and partial-observation formulations for multi-agent systems.
- Online adaptation for nonstationary dynamics and domain shifts.
These extensions aim to close the gap between elegant operator theory and robust real-world deployment.